A shoe store knows that the purchase amount by its customers is a random variable that follows a normal distribution with a mean of $30 and a standard deviation of $9. What is the probability that a randomly selected customer will spend $30 or more at this store?
Any customer who comes in is equally likely to spend over or under $30 at the store, since $30 is the mean of a normally distributed random variable.
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To find the probability that a randomly selected customer will spend $30 or more at the shoe store, we need to find the area under the normal distribution curve to the right of $30.
Step 1: Standardize the value.
We can standardize the value $30 using the formula:
z = (x - μ) / σ
where z is the z-score, x is the value we want to standardize, μ is the mean of the distribution, and σ is the standard deviation.
In this case, x = $30, μ = $30, and σ = $9.
Plugging in these values:
z = ($30 - $30) / $9
z = 0 / $9
z = 0
Step 2: Find the cumulative probability.
Now we need to find the cumulative probability associated with the z-score we calculated. We can use a standard normal distribution table or a calculator to determine this.
If we refer to a standard normal distribution table, we find that the cumulative probability corresponding to a z-score of 0 is 0.5000. This value represents the area under the curve to the left of the z-score.
Step 3: Subtract the cumulative probability from 1.
To find the area to the right of the z-score, we subtract the cumulative probability from 1.
Probability = 1 - 0.5000
Probability = 0.5000
Therefore, the probability that a randomly selected customer will spend $30 or more at this shoe store is 0.5000 or 50%.